Welcome to a lesson on how to determine volume in cubic feet and cubic yards.
We're given, a basement for a thirty foot by forty foot house is to be dug at a depth of
seven feet.
We are asked, how many cubic yards of earth need to be hauled away? Assume the
surface of the ground is level. We can model the amount of earth that needs to
be hauled away using this rectangular solid, where the dimensions are thirty feet by
forty feet by seven feet.
We will first determine the volume in cubic feed using the volume formula.
Volume equals length times width, times height. Then we will convert this volume
to cubic yards.
So the volume in cubic feet is equal to thirty feet
times forty feet,
times seven feet.
The volume is equal to eight thousand, four hundred cubic feet.
And now to answer the question, we convert eight thousand, four hundred cubic
feet to cubic yards.
First we write this volume as a fraction with a denominator of one.
Next we multiply it by a unit fraction.
Then we'll convert cubic feet to cubic yards.
Because we have cubic units here,
we will need three conversion factors using the conversion, one yard equals three feet,
in order for feet cubed, or cubic feet to simplify out. To begin building the
unit fraction, we know we must have feet in the denominator
and therefore, we have yards in the numerator.
Because the conversion is one yard equals three feet, one conversion factor is one
yard divided by three feet. But notice how this will not simplify out feet cubed,
this will only simplify out one factor of feet. We need two more of these
conversion factors to have a total of three conversion factors, so that we have
feet cubed in the denominator.
Notice by forming the unit fraction with three conversion factors, we now
have feet cubed in the denominator as well as the numerator. So the units of
feet cubed simplify to one.
Also notice how we're left with yards times yards, times yards, which does give
us the correct units of cubic yards. Multiplying, we have eight thousand, four
hundred divided by three times three, times three, which equals twenty-seven. Notice we have
an improper fraction.
This does simplify because eight thousand, four hundred and twenty-seven share a common factor of three. Let's first simplify
this by dividing the numerator and denominator by three.
This gives us a simplified improper fraction of two thousand, eight hundred divided by nine cubic yards.
Let's also express this as a mixed number. To convert the improper fraction
to a mixed number, we divide.
So we have two thousand, eight hundred divided by nine.
We first determine how many nines in twenty-eight, which is three. Three times nine is twenty-seven, subtract, the difference is
one, bring down the next digit.
Determine how many nines in ten, which is one. One times nine is nine, we subtract, the difference is one.
Bring down the next zero. Determine how many nines in ten, which is one. One times nine is nine,
subtract, the differences is one. So because the remainder is one, the quotient is three hundred eleven and
one-ninth.
Which means, eight thousand four hundred cubic feet is equal to three hundred eleven and one-ninth
cubic yards.
I hope you found this helpful.